John K Jordan
I agree with David's comment about the speed. If I'm imagining correctly what you are doing (I know nothing about such bowls), I think the speed should be about what you'd turn a platter or plate. Faster is generally cleaner, all else equal.
I suspect one issue will be trying to get smooth edges. Even a sharp fluted parting tool (I use two) will tear the end grain, but not as much as some other types of parting tools. One thing that might help is parting away from your line then sneak up on it, removing the tiniest bit of wood each time. I have had good luck with properly sharpened scrapers in cutting across end grain. I don't quite understand what you are doing and why the cut needs to be cleaned - I'm imagining these thin circles will be glued up then turned again, right. Maybe you could post some pictures or a diagram. (or get ideas from someone who knows what that kind of bowl is!)
One way people cut clean circles is with a router and circle jig. home made or otherwise. Or a router and templates.